Free energy of the edge of an open lipid bilayer basedon the interactions of its constituent molecules

Meisam Asgari n, Aisa BiriaDepartment of Mechanical Engineering, McGill University, Montréal, Canada, QC H3A 0C3

a r t i c l e i n f o

Article history:Received 27 November 2014Received in revised form6 April 2015Accepted 3 June 2015Available online 11 June 2015

Keywords:Free energyElasticityOpen lipid bilayerDifferential geometryBoundary curve of a surfaceMolecular interactions

a b s t r a c t

Lipid bilayers are the fundamental constituents of the walls of most living cells and lipid vesicles, givingthem shape and compartment. The formation and growing of pores in a lipid bilayer have attractedconsiderable attention from an energetic point of view in recent years. Such pores permit targeteddelivery of drugs and genes to the cell, and regulate the concentration of various molecules within thecell. The formation of such pores is caused by various reasons such as changes in cell environment,mechanical stress or thermal fluctuations. Understanding the energy and elastic behaviour of a lipid-bilayer edge is crucial for controlling the formation and growth of such pores. In the present work, theinteractions in the molecular level are used to obtain the free energy of the edge of an open lipid bilayer.The resulted free-energy density includes terms associated with flexural and torsional energies of theedge, in addition to a line-tension contribution. The line tension, elastic moduli, and spontaneous normaland geodesic curvatures of the edge are obtained as functions of molecular distribution, moleculardimensions, cutoff distance, and the interaction strength. These parameters are further analyzed byimplementing a soft-core interaction potential in the microphysical model. The dependence of theelastic free-energy of the edge to the size of the pore is reinvestigated through an illustrative example,and the results are found to be in agreement with the previous observations.

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1. Introduction

A phospholipid molecule consists of a hydrophilic head and twohydrophobic fatty-acid tails [1]. When suspended in an aqueoussolution at sufficient concentrations, phospholipid molecules self-assemble into structures such as lipid bilayers, in order to shield thetail groups from the solvent [2,3]. Lipid bilayers are the mainconstituents of cell membrane in most living organisms, as well asmodel membranes such as liposomes [4]. They provide the cell and itssubstructures with compartment and shape, and further, function asbarriers for water-soluble molecules such as water, ions, and proteins[5,6]. Lipid bilayers are composed of two adjacent leaflets of phos-pholipid molecules oriented transversely and set tail-to-tail.

Forming of open edges in lipid membranes results in theexposure of the tail groups at the edge to water [4], which isenergetically unfavourable. As a result, phospholipid moleculesrapidly rearrange around the exposed edge, forming a semicylind-rical rim along it. This rearrangement is the source of a line energyat the edge. In order to eliminate this edge energy, lipid bilayerscommonly tend to form closed structures such as spheroids [7].Nevertheless, they can transiently open due to various stimuli such

as mechanical stresses and thermal instabilities. The formation ofthese transient pores is essential for regulation of PH, transmem-brane electrochemical potential, and concentrations of differentmolecules in the cell [5]. Additionally, transient open membranesare formed during electro-formation [8]. More recently, stabilizingpores and control over their size have been pursued by means ofelectric fields [9], sonication [10], and use of edge-active chemicalagents [11]. The rapid progress in these techniques has attractedincreasing attention to the study of the open lipid bilayers,including molecular dynamic simulations, as well as continuummechanical treatment and numerical investigations of the equili-brium configurations [12,13].

Theoretical studies of the equilibrium and stability of poredmembranes have mainly relied on constitutive assumptions forthe edge, which neglect its flexural and torsional elasticity. Forinstance, Boal and Rao [14], Capovilla et al. [15], and Tu and Ou-Yang [16,17] considered the edge energy of an open lipid bilayer asa given constant. Tu and Ou-Yang [18] considered dependence ofthe edge energy on its geometry, namely geodesic and normalcurvatures. Nevertheless, their assumptions on the form of the lineenergy have not been precisely justified.

May [19] obtained the line energy of a lipid bilayer edgethrough optimization of the lipid packing at the vicinity of theedge. He modeled the edge as a semicylindrical micelle, and tookthe free energy per molecule to depend upon the chain length of

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International Journal of Non-Linear Mechanics

http://dx.doi.org/10.1016/j.ijnonlinmec.2015.06.0010020-7462/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: meisam.asgari@mail.mcgill.ca (M. Asgari).

International Journal of Non-Linear Mechanics 76 (2015) 135–143

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the molecules, their cross-sectional area, and the strength of theinteractions of the molecules with each other and with thesurrounding solution. Although successful in obtaining the linetension that framework did not capture the bending and torsionalenergetics of the edge. The gap in the literature to successfullyrelate the macro-scale edge energy to its microstructure hasmotivated the current study.

The interactions between the constituent molecules of a materialmay be used to obtain the free-energy density function of thatmaterial. For instance, Keller and Merchant [20] have employed sucha microphysical approach to extract the internal energy, surfacetension, and bending energy of a liquid surface and to relate itsbending rigidity to the molecular density and interaction potential. Ina recent application of the work of Keller and Merchant [20], theCanham–Helfrich free-energy density for a lipid vesicle was derivedbased on microphysical considerations [21]. Using the same approach,a model for the elastic free-energy of wormlike micelles was derived[22]. In doing so, the surfactant molecules comprising the wormlikemicelle were assumed to have constant length, and thus, weremodeled by one-dimensional rigid rods. The resulted expression forthe free energy was found to be quadratic in the curvature and torsionof the centerline of the micelle [22].

The current study adopts the microphysical approach of Keller andMerchant [20] to investigate the elastic behaviour of the edge of a lipidbilayer. Following May [19] and motivated by previous studies [23–26], the edge is modeled as a semicylindrical surface. In addition, thephospholipid molecules comprising the edge are modeled as one-dimensional rigid rods of constant length, oriented perpendicular tothe centerline of the edge. The applied framework enables us toextract the form of the free energy and the flexural and torsionalmoduli of the edge, based on the intermolecular energetic interactionbetween phospholipid molecules.

To find the free-energy density of the edge at a position x, weaccount for the interactions between all phospholipid molecules onthe edge within a cutoff distance δ from the molecules at x. Weassume that the phospholipid molecules are perpendicular to thecenterline of the edge. Our derivation relies on Taylor series expansionswith respect to a dimensionless parameter ϱ≔δ=ℓ⪡1, where ℓ is acharacteristic size parameter of the edge, such as its length. For ℓ takenas the length of the edge (or equivalently, the perimeter of a pore), itcan be related to the thickness or the length of the constituentmolecules, if the density of the molecules along the edge and theiraspect ratios are provided. The net free-energy of the edge results fromintegrating the free-energy density ϕ over the centerline of the edge.

The paper is structured as follows. In Section 2, requiredmathematical definitions are presented. Modeling assumptionsfor the edge of an open lipid bilayer are synopsized in Section 3.Section 4 is concerned with the derivation of the free-energydensity of such an edge. In Section 5, the consequences of choosinga spheroidal-particle potential (Berne and Pechukas [27] and Gayand Berne [28]) are considered to obtain the material parameterspresent in the derived model. As an illustrative example, asimplified model for a pore on a lipid bilayer is given in Section6, and the parameters obtained in Section 5 are used to find thefree-energy of the pore as a function of its size. Finally, the keyfindings of the study are summarized and discussed in Section 7.Details of the various derivations are provided in the Appendix.

2. Differential geometry of the bounding curve of a surface

Consider a smooth, orientable, open surface S representing theopen lipid bilayer, with boundary C¼ ∂S, as depicted schematicallyin Fig. 1. Let

C¼ fx : x¼ xðsÞ;0rsrLg; ð1Þ

denote the arclength parametrization of the closed boundarycurve C. On denoting the differentiation with respect to thearclength s by a superposed dot, it follows that j _x j ¼ 1, and thus,_x � €x ¼ 0 and j _x � €x j ¼ j €x j : ð2ÞThe unit tangent of C is introduced, in terms of the arclengthparametrization x, by

t≔ _x: ð3ÞSince the unit tangent t has a constant length, its arclengthderivative _t ¼ dt=ds is perpendicular to it, and thus, perpendicularto the curve C. The orientation of _t is called the unit normal of C,and is denoted by N. The curvature vector κ at any point of C isthen defined by the arclength derivative of the unit tangent t as

κ≔_t ¼ κN; ð4Þwhere κ denotes the magnitude of the curvature of C at that point,which is given in terms of the arclength parametrization x by

κ ¼ j _x � €x j ¼ j €x j : ð5ÞFor an arbitrary point on curve C at which κa0, the unit binormalvector is defined by B¼ t�N. The unit tangent t, unit normal N,and unit binormal B at each point of C, form the Frenet frameft;N;Bg at that point.

The torsion τ of C is defined by _B ¼ �τN, and is expressed interms of the arclength parametrization x as

τ¼ _x � ð €x � x…Þ

j €xj2 : ð6Þ

The torsion τ of C describes the tendency of the curve C to moveout of its osculating plane at a given point, or, equivalently, itmeasures the turnaround of the unit binormal B of C at a givenpoint. In general, a space curve is determined up to a rigidtranslation, by its two locally invariant quantities: the curvatureκ and torsion τ, both in terms of the arclength parameter s.

On the boundary curve C of the surface S, the unit normal tothe surface is denoted by n. Also, since _x is a unimodular vector, itsarclength derivative €x is perpendicular to _x, and thus, can beconsidered as a linear combination

€x ¼ κnnþκgn� _x; ð7Þ

Fig. 1. Mathematical identification of an open lipid bilayer as an open surface Swith boundary C¼ ∂S on which a Darboux frame has been shown. Also theschematic arrangements of phospholipid molecules in an interior point on S andat the vicinity of the edge C are depicted at a point.

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143136

of the unit normal n, and the product n� _x. Notice that the unitnormal n to the surface S is different from the unit normal N of thecurve C. Further, let p denote the unit vector in the tangent planeof S perpendicular to the unit tangent t while pointing outward.We call p the unit tangent-normal. The set of unit normal n to thesurface S at C, unit tangent-normal p, and the unit tangent t,which is t ¼ n� p, form the oriented basis ft;n;pg on C, known asthe Darboux frame (Fig. 1). Considering that the normal N andbinormal B of the Frenet frame of C are also perpendicular to t,they both lie in the plane spanned by the normal n and tangent-normal p of the Darboux frame. Therefore, they are related to nand p by

N¼ ð cosψ Þn�ð sinψ Þp;B¼ ð sinψ Þnþð cosψ Þp;

)ð8Þ

where ψ denotes the angle between the unit normal N of the curveC and the unit normal n to the surface S. Following the proofprovided in Appendix A, derivatives of t, n, and p with respect tothe arclength s along C are expressed as_t_n_p

264

375¼

0 κ cosψ �κ sinψ�κ cosψ 0 τþ _ψκ sinψ �τ� _ψ 0

264

375

tnp

264

375: ð9Þ

The quantity

τg ¼ τþ _ψ ; ð10Þis called geodesic torsion of the curve C on S. This quantitydescribes the rate of the rotation of the tangent plane of thesurface S about the unit tangent to the curve C with respect to thearc length s [29]. Also, τg can be expressed alternatively as

τg ¼ _x � ðn� _nÞ: ð11ÞFurther, the curvature vector κ of the curve C on the surface S isthe sum of the normal curvature vector κn, and the tangential (orgeodesic) curvature vector κg , i.e.

κ¼ κnþκg : ð12ÞThe normal curvature vector κn is the projection of the curvaturevector κ along the normal n of the surface S. The geodesiccurvature κg is perpendicular to the unit normal n to the surface,and, thus, lies in the tangent plane of the surface S. Hence,κn ¼ ðκ � nÞn and κg ¼ n� ðκ � nÞ: ð13ÞAccording to (4), the curvature vector κ of C is κ¼ κN. Since ψdenotes the angle between N and n, the magnitude κn of thenormal curvature κn is

κn ¼ κ � n¼ κN � n¼ κ cosψ : ð14ÞThe magnitude κg of the tangential (or geodesic) curvature vectorκg is a bending invariant and is given by

κg ¼ �p � _t ¼ �κp � N¼ ðn� tÞ � _t : ð15ÞAccording to _t ¼ κN, andn¼ ð cosψ ÞNþð sinψ ÞB; ð16Þthe right-hand side of (15) results

κg ¼ κgðn� tÞ where κg ¼ κ sinψ : ð17ÞAs mentioned earlier, the geodesic curvature vector κg at any pointof the curve C on the surface S is the vectorial projection of thecurvature vector κ of the curve C into the tangent plane of thesurface S at that point. This quantity is an intrinsic property of thesurface, which reflects the deviation of the curve C from a geodesicon the surface S [30]. In general, for a geodesic, the geodesiccurvature κg at any point is zero. Further, for a geodesic, the unitnormal N of the curve C coincides with the unit normal n of the

surface S, or, equivalently, the osculating plane of C at each point isperpendicular to the tangent plane of the surface S at that point[29]. This means that the Darboux frame and the Frenet frame fora geodesic are the same at any point.

According to (10), and the right-hand sides of (14) and (17), thearclength derivatives of ft;n;pg in (9) take the form_t ¼ κnn�κgp;_n ¼ �κntþτgp;_p ¼ κgt�τgn:

9>=>; ð18Þ

3. Modeling assumptions

The phospholipid molecules comprising the edge are allocated in away that their hydrophilic parts lie on a thin semicylindrical surface asshown in Fig. 2 to form a core shielding the hydrophobic tails from thesurrounding solution. The centerline of the edge is denoted by aboundary curve C. The following assumptions, which are based on thepreviously reported observations [23,24,26,31,32], are considered tomodel the edge of an open lipid bilayer:

ðiÞ The phospholipid molecules comprising the edge are modeledas one-dimensional rigid rods of the same length a.

ðiiÞ The lipid molecules are assumed to be perpendicular to thecenterline C, residing in the plane spanned by the unit normaln and the unit tangent-normal p, as depicted in Fig. 2. Thisassumption is valid as long as the concentration of the lipidmolecules on the edge C is sufficiently high.

ðiiiÞ The phospholipid molecules at any cross-section of the edgehave uniform angular distribution.

ðivÞ The distribution of the phospholipid molecules at any pointalong C is denoted by the molecular density function Π40. Incontrast to the angular distribution, which is assumed to beuniform because of symmetry considerations, the moleculardistribution along C may be non-uniform as a result oflocalized curvature.

Consider a lipid molecule at the position corresponding to s onC with orientation θ measured counterclockwise from the corre-sponding tangent-normal pðsÞ, as depicted schematically in Fig. 2b.Let the director dðs;θÞ denote the orientation of this molecule. Bythe second assumption, such a director can be expressed as alinear combination

dðs;θÞ ¼ ð cosθÞpðsÞþð sin θÞnðsÞ; ð19Þ

Fig. 2. (a) The schematic of a section of the edge of an open bilayer, (b) cross-sections of the edge at positions xðsÞ and xðtÞ with Darboux frame ft;n;pg at thosepositions.

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143 137

where pðsÞ and nðsÞ denote the tangent-normal and the unitnormal (to S) at the position corresponding to s.

4. Derivation of the free-energy density

In this section, the free-energy density of the edge of an openlipid bilayer is derived by taking into account the interactionsbetween the molecules comprising the edge. To do so, a micro-physical approach is applied, guided by the work of Keller andMerchant [20].

Consider two molecules, with directors d and d0, locatedrespectively at positions x and x0 interior to C. Let the interactionenergy (encompassing steric, electrostatic, and other relevanteffects) between the molecules under consideration be denoted by

Ωðx; x0;d;d0Þ: ð20Þ

Following Keller and Merchant [20], we assume that the interac-tion energy between two molecules separated by more than afixed cutoff distance δ vanishes, in which case

Ωðx; x0;d;d0Þ ¼ 0 if jx�x0 j4δ: ð21Þ

In the present setting, the cutoff distance δ is required to be smallrelative to the characteristic length ℓ of the edge, so that adimensionless measure ϱ of cutoff distance obeys

ϱ≔δℓ⪡1: ð22Þ

Hereafter, we restrict attention to interaction energies Ω that areof the form (20) but are also frame indifferent [33]. It then followsthat Ωðx; x0;d;d0Þ may depend on the positions x and x0 and thedirectors d and d0 only through the length jx�x0 j of the vectorbetween x and x0, the dot products ðx�x0Þ � d and ðx�x0Þ � d0formed by the directors and that vector, and the dot product d �d0 formed by the directors. Like Keller and Merchant [20], weassume that the dependence of the interaction energy on thelength of the relative position vector is scaled by the ratio ϱdefined in (22) and, thus, that

Ωðx; x0;d;d0Þ ¼ 2 ~Ωðϱ�2r � r; r � d; r � d0;d � d0Þ; ð23Þ

with r¼ x�x0. The factor of two on the right-hand side of (23) isfor simplifying later calculations. Notice that Ω depends explicitlyon δ, whereas ~Ω does not. Consequently, (21)–(23) yield

~Ωðs2;ρ1;ρ2;ρ3Þ ¼ 0 if s4ℓ; ð24Þ

where

ρ1 ¼ sϱr̂ � d; ρ2 ¼ sϱr̂ � d0; ρ3 ¼ sϱd � d0; ð25Þ

with r̂ being the unit vector corresponding to the intermolecularvector r.

As a consequence of the foregoing discussion, the net free-energy ϕnet of the edge can be expressed as

ϕnet ¼Z L0

12

Z π=2�π=2

ωðs;θÞ dθ !

ΠðsÞ ds; ð26Þ

where

ωðs;θÞ ¼Z L0

Z π=2�π=2

Ω xðsÞ; xðtÞ;dðs;θÞ;dðt;ηÞ� �ΠðtÞ dη dt ð27Þis the free energy due to the interactions between the moleculewith director dðs;θÞ at xðsÞ with all other molecules and where afactor of one-half compensates for the double counting of inter-actions arising from integrating over both s and t from 0 to L. From

(26), the free-energy density ϕ at position xðsÞ on C is simply

ϕ¼ 12

Z π=2�π=2

ωðs;θÞ dθ !

ΠðsÞ: ð28Þ

The function Π denotes the density of the lipid molecules at anypoint of the curve C. Following the proof which relies on theTaylor series expansion of the integrand of (28) with respect toϱ up to the second derivative term, provided in Appendix B, (28)becomes

ϕ¼ k○þk1κ2gþk2κ2nþk3κgþk4κnþk5κnκgþk6τ2g ; ð29Þ

which includes a quadratic expression in terms of κg , κn, and τg. Also,k○ and the coefficients ki in (29) are provided in Appendix B. Noticethat k○ is the standard line energy of the edge—the part which isindependent of edge geometry—while the coefficients ki, i¼ 1;2;5;6represent the flexural and torsional rigidities of the edge. Up to thesecond derivative term of the Taylor expansion considered here, thederived model (29) contains the linear terms of the normal curvatureκn and the geodesic curvature κg of the boundary curve C, while itdoes not incorporate the linear term of the geodesic torsion τg of C.Considering a simplification of (29) in the form

ϕ¼ k�○þk1ðκg�κg○Þ2þk2ðκn�κn○Þ2þk5κnκgþk6τ2g ; ð30Þ

where

κg○ ¼ � k32k1; κn○ ¼ � k42k2

; and k�O ¼ kO �

k234k1

� k24

4k2; ð31Þ

it can be inferred that the remaining coefficients k3 and k4 are relatedto the spontaneous geodesic and normal curvatures κg○ and κn○ of C.This transpires that our model captures the spontaneous normal andgeodesic curvatures κn○ and κg○ of the edge, while a spontaneousgeodesic torsion is absent from the free energy. Further, (29) and (30)include the coupling of the normal curvature κn and the geodesiccurvature κg via the term κnκg . However, the couplings of the geodesictorsion τg with the normal and geodesic curvatures κn and κg areabsent. Our calculations show that the latter terms only appear byincluding higher order terms in the Taylor expansion, and hence, areof less significance. In addition, if the molecules have non-uniformangular distribution, i.e. the molecular distribution function Π isallowed to depend upon θ or η, the model would also include alinear term in geodesic torsion τg that would lead to the presence of aspontaneous geodesic torsion.

The first term k○ on the right-hand side of (30) is insensitive tothe shape of the boundary. Since the molecular distribution on theboundary has implicit dependence upon the ambient temperatureand concentration, these effects may be encompassed in k○ and inthe moduli k1–k6. The line tension k○ has been obtained throughexperiments and molecular dynamic simulations for various typesof lipid bilayers (see [23] for a comparison of different measure-ments). However, there is not enough literature on measurementof the remaining coefficients. By fitting our model to existingmeasurements of the line tension, the controlling parameters ofthe interaction potential can be evaluated, and further used toobtain the remaining coefficients in (29).

The derived model (29) can be simplified into the previouslypresented models for the free-energy of the edge. In particular, thegeneral form (29) provides a development to the theoreticalinvestigations of open lipid bilayers presented by Tu and Ou-Yang [12] and Guven et al. [31].

4.1. Total free-energy of the edge

The net free-energy ϕnet associated with the elasticity of theedge of the open lipid bilayer is simply obtained by integrating the

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143138

free-energy density ϕ in (30) over the centerline C of the edge by

ϕnet ¼Z L0ϕ ds: ð32Þ

5. Applying a concrete interaction potential

The interaction potential in the model developed in the pre-vious section was assumed to be a general function of four frame-indifferent arguments in terms of the intermolecular vector andthe orientation of phospholipid molecules. There are numerousconcrete models for such interaction potentials between axisym-metric particles, which are vastly employed for numerical simula-tions of liquid crystals and other similar systems. Our derivationgives rise to integral representations for elastic moduli k○–k6 of theedge. Substituting the general form ~Ω in (B.1) with an availableinteraction potential yields the material parameters k○–k6 appear-ing in (29).

One of the standard examples among such pair interactionpotentials is the spheroidal-particle model proposed by Berne andPechukas [27] and Gay and Berne [28], in which the molecules areapproximated by ellipsoids of revolution, or spheroids (see Fig. 3).According to such model, the interaction potential between twomolecules with the intermolecular vector r and the directors d ande possesses the multiplicative decomposition

~Ωðr;d; eÞ ¼ ξðr̂ ;d; eÞζðr;d; eÞ; ð33Þwhere ξðr̂ ;d; eÞ and ζðr;d; eÞ denote the strength and distanceparameters respectively. The strength parameter ξðr̂ ;d; eÞ dependsupon the orientation of the molecules and that of the intermole-cular vector r through [28]

ξðr̂ ;d; eÞ ¼ 4ξ○ð1�χ2ðd � eÞ2Þν=2

1�χ0

2ðr̂ � dþ r̂ � eÞ21þχ 0d � e þ

ðr̂ � d� r̂ � eÞ21�χ 0d � e

!" #μ; ð34Þ

with ν, μ, and ξ○ being the fitting parameters to be chosen. Morespecifically, ν depends upon the arrangement type of the mole-cules (e.g. side-to-side or end-to-end), whereas ξ○ is a constantthat specifies the kind of molecules under consideration. Theparameter χ in (34) is the shape anisotropy parameter, given interms of the ratio of the length σe to the breadth σs of themolecules by

χ ¼ ðσe=σsÞ2�1

ðσe=σsÞ2þ1: ð35Þ

Also, the parameter χ 0 in (34) is given by

χ 0 ¼ ðεe=εsÞ1=μ�1

ðεe=εsÞ1=μþ1; ð36Þ

where εe and εs denote the strength parameters for end-to-endand side-to-side arrangement of the molecules respectively. The

distance parameter ζðr;d; eÞ in (33) is given by [27]

ζðr;d; eÞ ¼ exp �j rj2

ς2ðr̂ ;d; eÞ

� �; ð37Þ

where ςðr̂ ;d; eÞ is called the range parameter, and is given as afunction of the orientation of the molecules and that of theintermolecular vector r by

ςðr̂ ;d; eÞ ¼ σ○ 1�χ2

ðr̂ � dþ r̂ � eÞ21þχd � e þ

ðr̂ � d� r̂ � eÞ21�χd � e

!" #�1=2: ð38Þ

In (38), σ○ is related to the breadth of the molecules, σs, viaσ○ ¼

ffiffiffi2

pσs. Following Whitehead et al. [34], the parameters μ and

ν are chosen as

ν¼ �1 and μ¼ 2: ð39Þ

By applying the interaction potential (33) in (B.7), and assum-ing a constant molecular density Π along the boundary C, thecoefficients ki in (29) are obtained as

k○Π2ξ○σ○

¼ ffiffiffiffiπp erfðxÞI○;k1

Π2ξ○σ3○¼ k2Π2ξ○σ3○

¼ffiffiffiffiπ

p

128erfðxÞ χ2Jþðχ2þ2ÞI� �

þxe� x2

192χ2ðJþ IÞð2x2�3Þ�2Ið2x2þ3Þ� �;

k3 ¼ k4 ¼ k5 ¼ k6 ¼ 0; ð40Þ

where x is a dimensionless parameter defined as the ratio of thecut-off distance δ to σ○ as

x¼ δσ○

; ð41Þ

and I○, I and J are integral representations shown in Appendix C.Hence, the free-energy density ϕ in (29) specializes to

ϕ¼ k○þk1ðκ2nþκ2nÞ ¼ k○þk1κ2: ð42Þ

A single phospholipid molecule can be envisioned as a mole-cule in which a water-soluble spherical head is attached to a pairof water-insoluble tails. Here, we rely on the dimensions of aspecific kind of phospholipid molecule (DPPC/Water system)reported by Mashaghi et al. [35]. According to their estimation,the length of the aforementioned phospholipid molecule from thecenter of the head-group to the tail is � 22.5–30 Å, and thediameter of the head-group is � 7–10 Å. The total volume of thatmolecule is thus the sum of the volume of the spherical head andthat of a cylindrical tail-group. Based on the equality of thevolumes of the phospholipid molecule and that of the spheroidalreplacement, the aspect ratio σe=σs of the spheroid in Fig. 3 isobtained between 3 and 4. Hence, the schematic of the constantline energy k○ and that of the flexural rigidity k1 are depicted interms of the dimensionless cutoff distance δ=σ○ in Fig. 4, for themolecular aspect ratio σe=σs between 1 and 5. According to Fig. 4,the change in the constant line energy k○ and flexural rigidity k1are negligible after some value of the dimensionless cutoffdistance δ=σ○. Therefore, a rather conservative choice for thecut-off distance, which guarantees inclusion of all significantmolecular interactions, is

δn ¼ 3σ○: ð43Þ

This result has been used to obtain the constant part of the lineenergy k○ and flexural rigidity k1 of the edge in terms of themolecular aspect ratio, as depicted in Fig. 5.

Fig. 3. The schematic of a phospholipid molecule modeled as an ellipsoidalparticle.

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143 139

6. Illustrative example: dependence of free-energy on the poresize

In order to estimate the change of energy of a pore with its size,consider the simple case of a spherical lipid bilayer with radius R,with a pore of radius r at a distance h from its center, as depicted inFig. 6. For such a pore, the total curvature of the boundary curve Cis 1=r, and the geodesic torsion τg vanishes. Also, the normal andgeodesic curvatures find the forms

κn ¼1R

and κg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκ2�κ2n

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

prR

: ð44Þ

As a result, the free-energy density (30) specializes to

ϕ¼ k○þk1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

prR

�κg○ !2

þk21R�κn○

� �2þk5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

prR2

!;

ð45Þwhich with (32), yields a representation for the net free-energyϕnet of the pore as

ϕnet2πr

¼ k○þk1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

prR

�κg○ !2

þk21R�κn○

� �2þk5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�r2

prR2

!:

ð46ÞIt was demonstrated in the previous section that when the

spheroidal interaction potential [27] is employed in the micro-physical model, the spontaneous curvatures κg○ and κn○, and thecoefficient k5 vanish, while the bending moduli k1 and k2 find thesame value. Using those results in (46) yields

ϕnet2π

¼ k○rþk1r; ð47Þ

whereby the energy of the pore does not depend on the size of thelipid bilayer, nor on the placing of the pore on it. For the cut-offdistance δn ¼ 3σ○ and the molecular aspect ratios σe=σs ¼ 3 andσe=σs ¼ 4, the dependence of the net free-energy to pore size hasbeen demonstrated in Fig. 7. The second term on the right-handside of (47) leads to a minimum point for the free energy forroσ○, which does not fall in the physically relevant ranges of thepore size. For reasonable values of r=σ○, the first term on the right

Fig. 4. (a) Schematic of the constant line energy k○=Π2ξ○σ○ in terms of the

dimensionless cut-off distance δ=σ○ . (b) Schematic of the flexural rigidityk1=Π

2ξ○σ3○ in terms of the dimensionless cut-off distance δ=σ○ . As is evident fromthe plots, the change in k○=Π

2ξ○σ○ and k1=Π2ξ○σ3○ is negligible after δ¼ 3σ○ .

Consequently, the effective cut-off distance after which the potential decays rapidlycan be reasonably approximated by δ¼ 3σ○ (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. (a) Schematic of the constant line energy k○=Π2ξ○σ○ in terms of the aspectratio σe=σs for a cut-off distance δ=σ○ ¼ 3. (b) Schematic of the flexural rigidityk1=Π2ξ○σ○ in terms of the aspect ratio σe=σs for the cut-off distance δ¼ 3σ○ .

Fig. 6. Schematic of a pore on a spheroidal lipid bilayer.

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143140

of (47) is dominant and the dependence of the net energy on thepore size is effectively linear.

7. Discussion and summary

An expression for the free-energy density of the edge of anopen lipid bilayer was derived taking into account the interactionbetween the constituent molecules. The resulting expressioncontains quadratic terms in geodesic curvature, normal curvature,and geodesic torsion of the boundary curve and a term includingthe multiplication of geodesic and normal curvatures. The derivedfree-energy of the edge is the evidence of an excess energy due tothe specific arrangement of the phospholipid molecules in thevicinity of the boundary of an open bilayer, in accord with theresults of the existing molecular dynamic simulations [23–26].Further, our study supplements the previous molecular dynamicsimulations [23–26] and theories [19] in which the free energywas obtained as a constant, by providing the contribution due tobending and torsional energies. For certain classes of lipid bilayers,the bending free-energy, which can be captured by our frame-work, is of more importance in contrast to the line energy [36,37].In addition, our microphysical model justifies the constitutiveassumptions that appear in continuum mechanical theories foropen lipid bilayers [12,38,31].

Our derivation gives rise to integral representations for thematerial parameters present in the model. Specifically, the mole-cular origins of the spontaneous curvatures of the edge of a lipidbilayer have been investigated. A concrete soft-core interactionpotential for axisymmetric rod-like molecules was applied on thederived model to obtain those material parameters. Hence, aspecial form of the interaction potential suggested by Berne andPechukas [27] was employed to further explore the developedmicrophysical model. Assuming that the molecules are uniformlydistributed along the edge (i.e. Π ¼ constant), the spontaneouscurvatures and the torsional contribution to the energy vanish,resulting in (42), which includes only a constant part k○, which canbe interpreted as a line tension, and a contribution due to bendingwith flexural rigidity k1 ¼ k2. The dependence of the parameters k○and k1 on the aspect ratio σe=σs and the dimensionless cutoffdistance δ=σ○, for various aspect ratios common for phospholipidmolecules comprising lipid membranes, was investigated. It wasconcluded that increasing the cut-off distance after a valueδn ¼ 3σ○ does not affect those parameters. This result and thedefinition σ○ ¼

ffiffiffi2

pσs reveals that each molecule on the edge

interacts with less than 6 molecules in its vicinity along the edge.In view of this observation, the cut-off distance was set to δn tostudy the dependence of the line tension k○ and flexural rigidity k1

on the molecular aspect ratio σe=σs. It is evident from Fig. 4 thatthe line tension k○ is not as sensitive as the flexural rigidity k1 tothe molecular aspect ratio σe=σs. This difference can be quantifiedby considering the relative change of those quantities associatedwith the same increment of the molecular aspect ratio σe=σs.Further, with an increase in the aspect ratio, the line tensiondecreases while the flexural rigidity increases. These signals thatfor molecules with greater aspect ratios, inclusion of the bendingcontribution to the free-energy is of more significance. Consider-ing that a rod with a larger diameter shows more resistance tobending, the observation that the flexural rigidity of the edge isgreater for larger molecular aspect ratios (longer molecules lead tolarger cross-sectional diameter of the edge) agrees with whatwould be expected intuitively.

The energy functional obtained in (30) was used to exploreenergy of the degenerate case of a circular pore on a spheroidallipid bilayer, resulting in the expression (47). The pore size needsto be greater than the mean distance of molecules σ○. For suchsizes, the contribution of the flexural part to the energy given by(47) is negligible, and effectively, the energy increases linearlywith the pore size. Inspired by earlier investigations on rupturingof soap films [39], Litster [40] developed a continuum model forthe free-energy ϕ needed for opening-up of a pore in a lipidmembrane in the form

ϕ2π

¼Γr�12γsr

2; ð48Þ

with Γ being the line energy of the edge and γs the interfacialsurface tension. It can be inferred from (48) that transient poreswith sizes less than a critical radius rn ¼Γ=γs tend to reseal, whilethose having the size exceeding this critical radius, might growindefinitely, leading to the rupture of the membrane [41]. In otherwords, the energy to form a pore of radius r is determined by abalance between two competitive contributions: the energyrequired to create the edge of the pore, and the energy releasedby the pore surface [42]. Nevertheless, it is a common knowledgetoday (see for example the review by Jähnig [43]) that lipidmembranes possess zero surface tension, by which the secondterm on the right hand side of (48) vanishes. This transpires theagreement of the current result based on molecular interactions,with that obtained previously on continuum grounds. The increas-ing cost of generating a larger pore confirms the stability of a lipidmembrane with respect to the fluctuations that might bring abouttransient pores. Furthermore, the growth of stable pores inhomogenous lipid bilayers is only possible in the presence ofexternal stimuli such as an electric field.

The class of the interaction potentials (23) selected in thepresent study and the integration (26) only accounts for the lipidbilayers in which the physiochemical properties of the constituentmolecules are identical. An important corollary of our modelwould follow from a generalization of the arguments of theinteraction potential (23) and the integration (26), to allow usfor the interactions between phospholipid molecules of differentphysiochemical properties. Such a generalization permits model-ing perforated mixed lipid bilayers, such as those reported byOglȩcka et al. [44] and Jiang and Kindt [36]. Further, our modelaccounts for the elastic free-energy of the edge of open lipidbilayers in which the lipid molecules are tilted only at the edge,forming a semicylindrical rim along it. Another generalization ofthe present work would include tilt fields of smaller gradient, suchas those considered by Hamm and Kozlov [45], Rangamani andSteigmann [46], and Rangamani et al. [47]. In such an approach,the gradual tilt at the vicinity of the edge changes the thickness ofthe lipid bilayer and, thus, leads to the deviation of the conforma-tion of the edge from a semicylindrical shape. These potentialgeneralizations remain to be investigated in future.

Fig. 7. Schematic of the net free-energy ϕnet=Π2ξ○σ○ versus the scaled pore-sizer=σ○ for two values of the aspect ratio σe=σs and for the scaled cutoff distanceδ¼ 3σ○ .

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143 141

Acknowledgments

Financial support from National Institutes of Health (NIDCD)Grant DC 005788 (PI: Luc Mongeau) is gratefully acknowledged.The authors thank Mohsen Maleki for Figs. 1–3.

Appendix A. Detailed derivation of (18)

Let ψ denote the angle between the unit normal N to the curveC and the unit normal n to the surface S. The Darboux frameft;n;pg at any point on C is obtained by rotating the Frenet frameft;N;Bg about the unit tangent t by that angle. Hence,tnp

264

375¼

1 0 00 cosψ sinψ0 � sinψ cosψ

264

375

tNB

264

375: ðA:1Þ

Let the array X¼ t n p½ �T denote the Darboux frame, andY¼ t N B½ �T denote the Frenet frame at a given point on C, whereT denotes the transpose. Also let

Q≔

1 0 00 cosψ sinψ0 � sinψ cosψ

264

375 ðA:2Þ

denote the transformation between the two frames. Thus,

X¼QY or Y¼Q TX: ðA:3ÞThe derivative of the Frenet frame ft;N;Bg of C with respect to thearclength s follows from the Frenet–Serret formulas [48]

_t_N_B

264

375¼

0 κ 0�κ 0 τ0 �τ 0

264

375

tNB

264

375: ðA:4Þ

Thus,

_Y ¼AY¼ AQ TX; ðA:5Þwhere

A¼0 κ 0�κ 0 τ0 �τ 0

264

375: ðA:6Þ

Taking arclength derivative from both sides of (A.3)1 yields

_X ¼ _QYþQ _Y ; ðA:7Þwhere

_Q ¼0 0 00 � _ψ sinψ _ψ cosψ0 � _ψ cosψ � _ψ sinψ

264

375: ðA:8Þ

In view of (A.3)2 and the right-hand side of (A.5), (A.7) takes theform

_X ¼ ð _QQ T þQAQ T ÞX; ðA:9Þor, equivalently,

_t_n_p

264

375¼

0 κ cosψ �κ sinψ�κ cosψ 0 τþ _ψκ sinψ �τ� _ψ 0

264

375

tnp

264

375: ðA:10Þ

In view of κn ¼ κ cosψ , κg ¼ κ sinψ , and τg ¼ τþ _ψ , (A.10) simpli-fies to (18).

Appendix B. Derivation of the free-energy density (29)

In this Appendix, the expansion of (28) and (29) is presented.As mentioned earlier, only the molecules separated by a distanceless than δ may interact. Hence, the domain of the integral withrespect to t in (27) is replaced by ½�δ; δ�. Upon replacing ωðs;θÞ in(28), and substituting Ω by its equivalent form (23), and applyingthe change of variable t�t○ ¼ sϱ, (28) takes the form

ϕ¼ ϱZ ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

~Ω ϱ�2r � r; r � dðt○;θÞ;�

r � dðt○þsϱ;ηÞ;dðt○;θÞ � dðt○þsϱ;ηÞ�

Πðt○ÞΠðt○þsϱÞ dη dθ ds: ðB:1ÞIt is necessary to expand the right-hand side of (B.1) in powers of ϱneglecting terms of oðϱ2Þ. Introducing the abbreviationsn≔nð0Þ; t≔tð0Þ; p≔pðt○Þ;Π≔Πð0Þ; _Π≔ _Π ð0Þ; €Π≔ €Π ð0Þ;

)ðB:2Þ

and applying the identities given in (18), the following expansionsup to ϱ2 are obtained:

xðsϱÞ ¼ xð0Þþϱstþ12s2ϱ2κnn�

12s2ϱ2κgpþoðϱ4Þ;

nðsϱÞ ¼ s2ϱ2

2_κn�κgτg� ��sϱκn

� �tþ 1�s

2ϱ2

2κ2nþτ2g� �� �

n

þ s2ϱ2

2κnκgþ _τg� �þsϱτg

� �pþoðϱ2Þ;

pðsϱÞ ¼ s2ϱ2

2κnκg� _τg� ��sϱτg

� �nþ ϱsκgþ

s2ϱ2

2_κgþτgκn� �� �

t

þ 1�s2ϱ2

2τ2gþκ2g� �� �

pþoðϱ2Þ: ðB:3Þ

Therefore, the arguments of the interaction potential ~Ω in theright-hand side of (B.1) become

ϱ�2 jxðt○Þ�xðtϱÞj 2 ¼ s2þA1ϱ2s4þoðϱ2Þ;xðt○Þ�xðtϱÞ� � � dðt○;θÞ ¼ A2ϱ2s2þoðϱ2Þ;xðt○Þ�xðtϱÞ� � � dðtϱ;ηÞ ¼ A3ϱ2s2þoðϱ2Þ;dðt○;θÞ � dðtϱ;ηÞ ¼ A4þA5ϱsþA6ϱ2s2þoðϱ2Þ; ðB:4Þ

where tϱ ¼ t○þsϱ, and

A1 ¼ �κ2

12; A2 ¼

κg cosθ�κn sinθ2

; A3 ¼κn sinη�κg cosη

2;

A4 ¼ cos ðθ�ηÞ; A5 ¼ �τg sin ðθ�ηÞ;

A6 ¼κnκg2

sin ðθþηÞ� _τg2sin ðθ�ηÞ�τ

2g

2cos ðθ�ηÞ

�κ2g

2cosθ cos η�κ

2n

2sin θ sinη: ðB:5Þ

Expanding ~Ω up to ϱ2 and using

ΠðsϱÞ ¼Πþsϱ _Πþs2ϱ2

2€Πþoðϱ2Þ; ðB:6Þ

result in the following energy-density for the edge:

ϕ¼Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

ϱΩðs;θ;ηÞΠ2 dη dθ dsþ12ϱ3Ωðs;θ;ηÞs2Π €Π

dη dθ ds

þκ2g ϱ3Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

�s2Π212

Ω1ðs;θ; ηÞs2þ6Ω4ðs;θ; ηÞ cos θ cos η� �

dη dθ ds

( )

þκ2n ϱ3Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

�s2Π212

Ω1ðs;θ;ηÞs2�(

þ6Ω4ðs;θ;ηÞ sin θ sinη�dη dθ dsg

M. Asgari, A. Biria / International Journal of Non-Linear Mechanics 76 (2015) 135–143142

þκg ϱ3Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

s2Π2

2Ω2ðs;θ;ηÞ cosθ�Ω3ðs;θ;ηÞ cosη� �

dη dθ ds

( )

þκn ϱ3Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

s2Π2

2Ω3ðs;θ;ηÞ sinη�Ω2ðs;θ;ηÞ sin θ� �

dη dθ ds

( )

þτ2g ϱ3Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

s2Π2

2Ω44ðs;θ;ηÞ sin 2ðθ�ηÞ�(

�Ω4ðs;θ;ηÞ cos ðθ�ηÞ�dη dθ dsg

þκnκg ϱ3Z ℓ�ℓ

Z π=2�π=2

Z π=2�π=2

s2Π2

2Ω4ðs;θ;ηÞ sin ðθþηÞ dη dθ ds

( );

ðB:7Þwhere

Ωðs;θ;ηÞ≔ ~Ωðs2;0;0; cos ðθ�ηÞÞ;Ω iðs;θ;ηÞ≔ ~Ω ;iðs2;0;0; cos ðθ�ηÞÞ;Ω iiðs;θ;ηÞ≔ ~Ω;iiðs2;0;0; cos ðθ�ηÞÞ;

9>>=>>; iAf1;2;3;4g: ðB:8Þ

Note that Ω, Ω i and Ω ii vanish when s4ℓ. Thus, the finalexpression for the free-energy density of the edge becomes

ϕ¼ k○þk1κ2gþk2κ2nþk3κgþk4κnþk5κnκgþk6τ2g ; ðB:9Þor, equivalently,

ϕ¼ k�○þk1ðκg�κg○Þ2þk2ðκn�κn○Þ2þk5κnκgþk6τ2g ; ðB:10Þ

where the parameters k○–k6 are given by the integral representa-tions in (B.7).

Appendix C. Integral representations I○, I and J in (40)

I○ ¼Z π=2�π=2

Z π=2�π=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�χ2 cos 2ðθ�ηÞ

qdη dθ;

I ¼Z π=2�π=2

Z π=2�π=2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�χ2 cos 2ðθ�ηÞ

p dη dθ;J ¼

Z π=2�π=2

Z π=2�π=2

cos ð2θ�2ηÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�χ2 cos 2ðθ�ηÞ

p dη dθ: ðC:1Þ

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Free energy of the edge of an open lipid bilayer based on the interactions of its constituent moleculesIntroductionDifferential geometry of the bounding curve of a surfaceModeling assumptionsDerivation of the free-energy densityTotal free-energy of the edge

Applying a concrete interaction potentialIllustrative example: dependence of free-energy on the pore sizeDiscussion and summaryAcknowledgmentsDetailed derivation of (18)Derivation of the free-energy density (29)Integral representations I○, I and J in (40)References